Download Many-body wave scattering by small bodies

arXiv:math-ph/0606067v2 18 Aug 2006
Many-body wave scattering by small bodies
A.G. Ramm
Mathematics Department, Kansas State University,
Manhattan, KS 66506-2602, USA
[email protected],
fax 785-532-0546, tel. 785-532-0580
http://www.math.ksu.edu/eramm
Abstract
Scattering problem by several bodies, small in comparison with the wavelength,
is reduced to linear algebraic systems of equations, in contrast to the usual reduction
to some integral equations.
1
Introduction
Acoustic or electromagnetic (EM) wave scattering by one or several bodies is usually
studied by reducing the problem to solving some integral equations. In this paper we
show that if the bodies are small in comparison with the wavelength, then the scattering
problem can be reduced to solving linear algebraic systems with matrices whose elements
have physical meaning. These elements are electrical capacitances or elements of electric
and magnetic polarizability tensors. The author has derived analytical explicit formulas
allowing one to calculate these quantities for bodies of arbitrary shapes with arbitrary
desired accuracy (see [1]).
We derive these linear algebraic systems and give formulas for the elements of the
matrices of these systems. There is a large literature on wave scattering by small bodies,
see [1] and references therein. The theory was originated by Lord Rayleigh [3], who
understood that the main term in the scattered field is the dipole radiation if the body
is small. Rayleigh did not give formulas for calculating the induced dipole moments
for small bodies of arbitrary shapes. The dipole moments are uniquely defined by the
MSC: 35J10, 35P25, 74J20, 78A40, 78A45
PACS: 0340K, 4110H, 4320
key words: many-body scattering, acoustic wave scattering, electromagnetic wave scattering, numerical analysis
polarizability tensors. Therefore, the formulas, derived by the author (see [1]), allow one
to calculate the dipole radiation for acoustic and EM wave scattering by small bodies of
arbitrary shapes.
2
Acoustic wave scattering by small bodies
Let us start with acoustic wave scattering. Consider the problem
(∆ + k 2 )u = 0 in R3 \ (
m
[
Dm )
(1)
Sm := ∂Dm
(2)
m=1
u |Sm = 0,
1 ≤ m ≤ M,
∂v
− ikv = o
∂r
u = u0 + v,
(3)
1
,
r
(4)
r := |x| → ∞,
where ∆ is the Laplacean, u0 is an incident field which solves equation (1). Often,
u0 = eikα·x , where α ∈ S 2 is a given vector and S 2 is the unit sphere.
Let us look for the solution of the form
u = u0 +
M Z
X
m=1
g(x, s)σm (s)ds,
g(x, y) :=
Sm
eik|x−y|
4π|x − y| ,
(5)
where σm , 1 ≤ m ≤ M, are to be chosen so that the boundary conditions (2) hold. The
function (5) satisfies (1) and (3)- (4) for any σm ∈ L2 (Sm ). The scattering amplitude is:
′
A(α , α) =
limx
|x|→∞,
|x|
−ik|x|
=α′
|x|e
Z
M
X
1
′
e−ikα ·s σm ds,
v=
4π Sm
m=1
α′ :=
x
.
|x|
(6)
Let
a := max diam Dm ,
1≤m≤M
and
d := min dist(Dm , Dj ).
m6=j
We assume
ka ≪ 1,
Then
a ≪ d.
′
e−ikα ·(s−xm ) ≈ 1 if
2
xm ∈ D,
(7)
so
Z
′
M
m
X
X
Qm −ikα′ ·xm
e−ikα ·xm
σm ds :=
A(α , α) =
e
,
4π
4π
S
m
m=1
m=1
′
Qm :=
Z
σm ds,
(8)
Sm
where xm ∈ Dm and α′ is defined in (6). Since Dm is small, it does not matter which
point xm one takes in Dm . The Qm plays the role of the total charge on the surface Sm .
If minm |x − xm | ≫ a and xm ∈ Dm , then
M
h
X
a i
eik|x−xm|
.
u(x) = u0 (x) +
Qm 1 + O ka +
4π|x − xm |
d
m=1
Let us derive a formula for Qm . Using the boundary condition (2), one gets:
Z
X
g(sm, s)σm (s)ds,
0 = u0 (sm ) +
g(sm , xj )Qj +
(9)
(10)
Sm
j6=m
where sm ∈ Sm .
Since ka ≪ 1, one has
g(sm , s) = g0 (sm , s) + O(ka),
where
g0 (s, t) :=
1
.
4π|x − t|
Therefore equation (10) is the equation for the electrostatic charge distribution σm on
the surface Sm of a perfect conductor Dm , charged to the potential
X
um := −u0 (sm ) −
g(sm , xj )Qj .
j6=m
The total charge on Sm is:
Qm = Cm Um ,
where Cm is the electrical capacitance of the conductor with the shape Dm . The total
charge is defined as:
Z
σm ds.
Qm :=
Sm
Therefore, one gets:
Qm = Cm −u0 (sm ) −
X
g(sm , xj )Qj
j6=m
!
,
1 ≤ m ≤ M,
(11)
where Cm is the electrical capacitance of the perfect conductor with the boundary Sm .
3
Linear algebraic system (11) allows one to find Qj , 1 ≤ j ≤ M. If
max
1≤m≤M
X
j6=m
Cm
< 1,
4π|sm − xj |
(12)
then the matrix of the system (11) has diagonally dominant elements and, consequently,
can be solved by iterations.
The approximate solution to the many-body scattering problem (1)–(4) is given by
formula (9), where Qm are determined from linear algebraic system (11).
Let us give a formula from [1] for the capacitance of a perfect conductor D with the
boundary S. Denote the area of S by |S|. We assume that the conductor is placed in the
medium with the dielectric permittivity ε0 = 1. In this case the approximate formula for
the capacitance is (see [1], p. 26):

−1




Z
 −1 n Z Z dsdt Z

(n)
2
C = 4π|S|
. . . ψ(t, t1 ) . . . ψ(tn−1 , tn )dt1 . . . dtn
(13)


2π
S S rst
S
S




| {z }
n
C
(0)
4π|S|2
≤ C,
=
J
∂ 1
,
ψ(t, s) =
∂Nt rst
times
J :=
,
Z Z
S
S
dsdt
,
rst
rst := |s − t|,
and the error estimate of formula (13) is:
|C (n) − C| = O(q n ),
0 < q < 1,
(14)
where q depends on the geometry of S, and n = 1, 2, 3..... is the approximation order.
If the boundary condition
uN = ζu on Sm
(15)
is imposed in place of the Dirichlet condition (2), and ζ is the impedance, then Cm in
(11) is replaced by
Cm
,
(16)
Cmζ :=
1 + Cm (ζ|S|)−1
see [1], p. 97.
If
uN |Sm = 0,
1 ≤ m ≤ M,
then the formula for the solution to problem (1), (17), (3), (4), is
"
#
M
M
X
X
∂u(xm ) (x − xm ),p
u(x) = u0 (x) +
q(x, xm )Vm ∆u(xm ) +
βpq,m ik
,
∂x
|x
−
x
|
m,q
m
m=1
p,q=1
4
(17)
(18)
where (x − xm ),p is the p-th coordinate of the vector x − xm , ∂x∂m,q is the derivative with
respect to the q-th coordinate of x calculated at the point xm , and βpq,m is the magnetic
polarizability tensor of Dm , defined by the formula ([1], p.98):
Z
Vm βpq,m =
sp σ(s)ds,
S
where Vm is the volume of Dm , the function σ solves the equation
σ = Aσ − 2Nq ,
N is the exterior unit normal to Sm , and
Z
1
∂
σ(t)dt,
Aσ =
Sm ∂N 2πrst
rst = |s − t|.
The formulas for the tensor βpq,m , analogous to the formulas (13)-(14) for the capacitance, are derived in [1, p.55, formula (5.15)]. The unknown quantities ∆u(xm ) and
∂u(xm )
, 1 ≤ m ≤ M, 1 ≤ q ≤ 3, in (18) can be found from the following linear algebraic
∂xq
system, analogous to (11):
∆u(xm ) = ∆u0 (xm ) − k
2
M
X
g(xm , xj )Vj [∆u(xj ) +
p,q=1
j6=m,j=1
M
X
∂u(xm )
∂u0 (xm )
=
+
∂xm,q
∂xm,q
j6=m,j=1
3
X
βpq,j ik
∂u(xj ) (xm − xj ),p
]
∂xj,q |xm − xj |
(19)
3
X
∂u(xj ) (xm − xj ),p
∂g(xm , xj )
Vj [∆u(xj ) +
βpq,j ik
].
∂xm,q
∂xj,q |xm − xj |
p,q=1
(20)
In (19) we have used the equation
∆g(x, y) = −k 2 g(x, y),
which holds if x 6= y.
From the linear algebraic system (19)–(20) one finds the unknowns ∆u(xm ) and
∂u(xm )
, 1 ≤ m ≤ M, 1 ≤ q ≤ 3.
∂xm,q
If conditions (7) hold, then system (19)–(20) has a unique solution which can be
obtained by iterations.
This completes the description of our method for solving many-body scattering problem for small bodies and acoustic (scalar) waves.
5
3
Electromagnetic wave scattering by small bodies
In the problem of electromagnetic (EM) wave scattering by many small bodies we assume
a ≪ λ ≪ d.
(21)
This assumption is more restrictive than (7). The reason is: in EM theory the fields
are obtained by an application of first order differential operators, for instance ∇×, to
potentials, such as the vector potential. Applying this operator and calculating the field
1
in the far zone one neglects the term | x−x
| compared with the term k. This means that
m
the following inequality is assumed:
1
1
≪ ,
d
λ
or
d ≫ λ.
R
In the acoustic wave theory the potential itself S g(x, s)σds has physical meaning, it is
the acoustic pressure, and this pressure is studied. Therefore, the condition d ≫ λ does
not appear.
Condition (7) allows one to have many small particles on the distance of order λ,
while condition (21), namely the inequality d ≫ λ, does not allow this. Recall that d is
the minimal distance between two neighboring particles. The formula for the scattering
amplitude, analogous to (8), for EM wave scattering by small bodies is (see [2]):
M
1 X
′
A(θ , θ) =
Sm Um e−ikθ ·xm .
4π m=1
′
(22)
E
Here U = H
is a 6-component vector, Sm is a 6x6 matrix, the scattering matrix, ε0 and
µ0 are dielectrical and magnetic parameters of the medium, in which the body Dm is
placed, and θ, θ′ are the unit vectors in the direction of the incident and scattered waves,
respectively. These vectors were denoted α and α′ in Section 2. We have changed the
notations because in EM theory α denotes the polarizability tensor.
The formula for S is (cf. [2])


3/2
µ0
′ ′
′ e
− 1/2 [θ , βH]
k 2 Vm αE − θ (θ , αE)
E

ε0
=
Sm
(23)
1


2
4π
H
ε0
′
′ ′ e
e
[θ , αE]
µ0 (β − θ (θ , βH))
µ0
.
Here Vm is the volume of Dm , α is the electric polarizability tensor of Dm , βe is the
magnetic polarizability tensor of Dm . In [1, pp. 54–55] the author derives analytical
formulas for calculation of the polarizability tensors α and β,
βe := α(e
γ ) + β,
β := αij (−1),
6
γ :=
e
µ − µ0
,
µ + µ0
γ :=
ε − ε0
.
ε + ε0
(24)
Tensors β and βe are expressed through the polarizability tensor α = α(γ). One has
γ = −1 if ε = 0. Here [·, ·] is the vector product, (·, ·) is the scalar product.
The analytic formula from [1], p. 54, formula (5.9), for the tensor α = αij (γ), 1 ≤ i,
j ≤ 3, that we referred to above, is analogous to formulas (13)–(14) for the electrical
capacitance. The incident direction θ enters via the vectors E and H, which depend
on θ. These vectors are calculated in formula (23) at the point xm . The values of these
vectors are determined from a linear algebraic system of equations. This system is derived
similarly to the derivation of the systems (11) and (19)–(20). We do not write down this
system since it would take much space, but the ideas are the same as the ones used in
the derivations of (11) and (19)–(20).
4
Conclusions
In this paper it is shown how to reduce rigorously the many-body scattering problem
to linear algebraic system in the case when the bodies are small in comparison with the
wavelength. The theory is constructed for acoustic and EM wave scattering. The basic
physical assumptions are (7) for acoustic scattering, and (21) for EM scattering.
References
[1] Ramm, A. G. , Wave scattering by small bodies of arbitrary shapes, World
Sci. Publishers, Singapore, 2005.
[2]
, Equations for the self-consistent field in random medium, Phys.Lett. A,
312, N3-4, (2003), 256-261.
[3] Rayleigh, J. , Scientific Papers, Cambridge, 1922.
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